Finite Group Databasebeta

← #13411789

Subgroups — #13411789

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739694C211yesmaximal, characteristic
2#11739604C211yesmaximal, characteristic
3C7×#1736C211yesmaximal, characteristic
4#11729749C211yesmaximal, characteristic
5#11729742C211yesmaximal, characteristic
6#11729330C211yesmaximal, characteristic
7#11729308C211yesmaximal, characteristic
8#11726597C211yesmaximal, characteristic, complemented
9#11726592C211yesmaximal, characteristic, complemented
10#11726582C211yesmaximal, characteristic, complemented
11#11726574C211yesmaximal, characteristic, complemented
12#11726569C211yesmaximal, characteristic, complemented
13#11726563C211yesmaximal, characteristic, complemented
14#11726555C211yesmaximal, characteristic, complemented
15#11726550C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×#6489—21no—
20C2×#6453C2211yescharacteristic
21#89984C2211yescharacteristic
22#89984C2211yescharacteristic
23#89983—21no—
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×#6391C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30#89757C2211yescharacteristic
31#89757C2211yescharacteristic
32#89757C2211yescharacteristic
33#89753—21no—
34#89753—21no—
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89721—21no—
42#89720—21no—
43#89705—21no—
44#89700—21no—
45C7×#340C2211yescharacteristic
46C7×#340C2211yescharacteristic
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89345C2211yescharacteristic
52#89345C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26014—71nomaximal, Sylow
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C2×C7×M16—41no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×#117—22no—
68C7×#117—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4Q814yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7D411yescharacteristic
81C2×C22⋊Dic7D411yescharacteristic
82#6489—44no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7C2311yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6453—22no—
102#6453—41no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113C22⋊(C7⋊C8)—22no—
114#6392—22no—
115#6391—41no—
116Dic7⋊C8—22no—
117Dic7⋊C8—41no—
118#6382—41no—
119#6381—22no—
120#2441—71no—
121#2344—71no—
122#2100—71no—
123#2088—71no—
124#2086—71no—
125#2059—71no—
126#1989—71no—
127#1948—71no—
128#1921—71no—
129#1918—71no—
130#1854—71no—
131#1851—71no—
132#1824—71no—
133#1809—71no—
134#1736D711yescharacteristic
135C24×C7C2×D411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—24no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×Q812yes—
149C22×C4×C7#4112yes—
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C7×M16—44no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8#4112yes—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4C2×Q812yes—
173C7×C4⋊C4C2×Q812yes—
174C7×C4⋊C4#4112yes—
175C7×C22⋊C4—42no—
176C42×C7—42no—
177C22⋊Dic7—22no—
178C22⋊Dic7—41no—
179C22⋊Dic7—42no—
180C22⋊Dic7—81no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—42no—
190C4⋊Dic7—42no—
191C4⋊Dic7—42no—
192C4⋊Dic7—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—42no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—42no—
202C4×Dic7—42no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C22×M16C2×D711yescharacteristic
210C2×#125—71no—
211C2×#117—141no—
212C2×#117C2×D711yescharacteristic
213C22×C4⋊C4C2×D711yescharacteristic
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8—71no—
220C2×C22⋊C8—71no—
221#399—141no—
222#396—141no—
223#389—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#389—141no—
228#387—141no—
229#387—141no—
230#387—141no—
231#384—71no—
232#381—141no—
233#380—71no—
234#380—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C22⋊C4—71no—
238#340—71no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340C2×D711yescharacteristic
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#340C2×D711yescharacteristic
250#336—71no—
251#336—71no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C22⋊Q812yes—
259C23×C7C22×D411yescharacteristic
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—48no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7C22⋊Q812yes—
280C2×C4×C7C22×Q811yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7H(5,𝔽2)11yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7AGL(1,ℤ/8)12yes—
310C7×C8—22no—
311C7×C8—41no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C23×C4—21no—
315C23×C4C22×D711yescharacteristic
316C2×M16—22no—
317C2×M16—22no—
318C2×M16—41no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323#125—142no—
324#125—281no—
325#117—22no—
326#117—41no—
327#117—284no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4Dic1414yes—
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—71no—
349C2×C22⋊C4—71no—
350C2×C22⋊C4—141no—
351C2×C22⋊C4—141no—
352C2×C42—21no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—141no—
356C2×C42—141no—
357#107—142no—
358#107—142no—
359#106—281no—
360#106—281no—
361C4⋊C8—142no—
362C4⋊C8—281no—
363C22⋊C8—142no—
364#95—71no—
365#95—71no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7#54711yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57811yescharacteristic
388C4×C7—22no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—81no—
393C4×C7#53311yescharacteristic
394C4×C7#54411yescharacteristic
395C4×C7#55111yescharacteristic
396C4×C7#55211yescharacteristic
397Dic7—22no—
398Dic7—41no—
399Dic7—41no—
400Dic7—41no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24D4×D711yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4C2×Dic1412yes—
426C22×C4D4×D711yescharacteristic
427C22×C4#115912yes—
428C22×C4C23×D711yescharacteristic
429M16—44no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8#115712yes—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—282no—
457C4⋊C4—282no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4C2×Dic1412yes—
465C4⋊C4C2×Dic1412yes—
466C4⋊C4#115712yes—
467C22⋊C4—42no—
468C22⋊C4—142no—
469C22⋊C4—281no—
470C22⋊C4—282no—
471C22⋊C4—561no—
472C42—42no—
473C42—141no—
474C42—141no—
475C42—141no—
476C42—141no—
477C42—282no—
478C42—282no—
479C2×C7—42no—
480C2×C7#243811yescharacteristic
481C2×C7#307711yescharacteristic
482C2×C7#308211yescharacteristic
483C2×C7#331711yescharacteristic
484C2×C7#333011yescharacteristic
485C2×C7#334511yescharacteristic
486C2×C7#335211yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#644112yes—
494C23C2×D4×D711yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—72no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4#644112yes—
539C2×C4C22×Dic1411yescharacteristic
540C2×C4C2×#115711yescharacteristic
541C2×C4C2×#115711yescharacteristic
542C2×C4C2×#115911yescharacteristic
543C2×C4#654811yescharacteristic
544C2×C4#655111yescharacteristic
545C8—22no—
546C8—41no—
547C8—281no—
548C8—281no—
549C7#2601411yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#644111yescharacteristic
558C22#9018111yescharacteristic
559C22#9019811yescharacteristic
560C22D7×AGL(1,ℤ/8)11yescharacteristic
561C22#9046411yescharacteristic
562C22#9046911yescharacteristic
563C22#9047011yescharacteristic
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—142no—
570C4—281no—
571C4—281no—
572C4—281no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9016111yescharacteristic
578C4#9018611yescharacteristic
579C4#9019111yescharacteristic
580C4#9020411yescharacteristic
581C2—42no—
582C2#1173079511yescharacteristic
583C2#1173634611yescharacteristic
584C2#1173635111yescharacteristic
585C2#1173636211yescharacteristic
586C2#1173639511yescharacteristic
587C2#1173667011yescharacteristic
588C2#1173667811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1736C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1736D7
hypocenterC7#26014
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #11726597C2semidirect
9. #11726592C2semidirect
10. #11726582C2semidirect
11. #11726574C2semidirect
12. #11726569C2semidirect
13. #11726563C2semidirect
14. #11726555C2semidirect
15. #11726550C2semidirect
549. C7#26014semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117396942
#117396042
C7×#17362
#117297492
#117297422
#117293302
#117293082
#117265972
#117265922
#117265822
#117265742
#117265692
#117265632
#117265552
#117265502
#260147